Remarks on quantiles and distortion risk measures

Jan Dhaene, Alexander Kukush, Daniël Linders, Qihe Tang

Research output: Contribution to journalArticlepeer-review


Distorted expectations can be expressed as weighted averages of quantiles. In this note, we show that this statement is essentially true, but that one has to be careful with the correct formulation of it. Furthermore, the proofs of the additivity property for distorted expectations of a comonotonic sum that appear in the literature often do not cover the case of a general distortion function. We present a straightforward proof for the general case, making use of the appropriate expressions for distorted expectations in terms of quantiles.

Original languageEnglish (US)
Pages (from-to)319-328
Number of pages10
JournalEuropean Actuarial Journal
Issue number2
StatePublished - Dec 1 2012
Externally publishedYes


  • Comonotonicity
  • Distorted expectation
  • Distortion risk measure
  • Quantile
  • TVaR

ASJC Scopus subject areas

  • Statistics and Probability
  • Economics and Econometrics
  • Statistics, Probability and Uncertainty

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